Multimode josephson parametric converter: coupling josephson ring modlator to metamaterial

ABSTRACT

A technique relates to operating a multimode josephson parametric converter as a multimode quantum limited amplifier. The multimode. Josephson parametric converter receives multiple quantum signals in parallel at different resonance frequencies. The multimode Josephson parametric converter amplifies simultaneously the multiple quantum signals, according to pump signals applied to the multimode Josephson parametric converter. The multiple quantum signals having been amplified at the different resonance frequencies are reflected, according to the pump signals.

PRIORITY

This application is a divisional of U.S. Non-Provisional ApplicationSer. No. 14/871,562, entitled “MULTIMODE JOSEPHSON PARAMETRIC CONVERTER:COUPLING JOSEPHSON RING MODULATOR TO METAMATERIAL”, filed Sep. 30, 2015,which is incorporated herein by reference in its entirety.

BACKGROUND

The present invention relates to quantum information processing in themicrowave domain using superconducting circuits, and more specifically,to a multimode Josephson parametric converter.

Recent progress in solid-state quantum information processing hasstimulated the search for amplifiers and frequency converters withquantum-limited performance in the microwave domain. Depending on thegain applied to the quadratures of a single spatial and temporal mode ofthe electromagnetic field, linear amplifiers can be classified into twocategories (phase sensitive and phase preserving) with fundamentallydifferent noise properties. Phase-sensitive amplifiers squeeze the inputnoise and signal in one quadrature of the microwave field at the expenseof inflating the noise and signal in the other quadrature without addingnoise of their own to the processed signal, but are useful only in casesin which the quantum information is encoded in one quadrature of themicrowave field. A phase-preserving amplifier on the other handamplifies both quadratures of the input noise and signal at the expenseof adding at least a noise equivalent to a half input photon at thesignal frequency. Such an amplifier would be useful in many quantumapplications, including qubit readout. One successful realization of anon-degenerate-intrinsically phase-preserving-superconducting parametricamplifier is based on a Josephson ring modulator, which consists of fourJosephson junctions in a Wheatstone bridge configuration. The devicesymmetry enhances the purity of the amplification process, i.e.,eliminates or minimizes certain undesired nonlinear processes, and alsosimplifies both its operation and its analysis.

SUMMARY

According to one embodiment, a method of operating a multimode Josephsonparametric converter as a multimode quantum limited amplifier isprovided. The method includes receiving, by the multimode Josephsonparametric converter, multiple quantum signals in parallel at differentresonance frequencies, and amplifying simultaneously, by the multimodeJosephson parametric converter, the multiple quantum signals, accordingto pump signals applied to the multimode Josephson parametric converter.Also, the method includes reflecting the multiple quantum signals havingbeen amplified at the different resonance frequencies, according to thepump signals.

According to one embodiment, a method of operating a multimode Josephsonparametric converter to generate multiple pairs of entangled photons.The method includes receiving, by a first multimode resonator in themultimode Josephson parametric converter, a first group of signals atdifferent resonance frequencies of the resonance modes of the firstmultimode resonator, where the first multimode resonator is a firstleft-handed transmission line. The method includes receiving, by asecond multimode resonator in the multimode Josephson parametricconverter, a second group of signals at different resonance frequenciesof the resonance modes of the second multimode resonator, where thesecond multimode resonator is a second left-handed transmission line.Also, the method includes receiving pump signals, by the secondmultimode resonator, where the pump signals are a first frequency sumthrough a last frequency sum, and generating pairs of entangled photons,the pairs of entangled photons including a first pair through a lastpair.

According to one embodiment, a method of remote entanglement of multiplequbits by measurement is provided. The method includes receiving, by afirst multimode resonator in the multimode Josephson parametricconverter, a first group of readout signals resonant with resonancemodes of the first multimode resonator, where the first multimoderesonator is a first left-handed transmission line. The method includesreceiving, by a second multimode resonator in the multimode Josephsonparametric converter, a second group of readout signals resonant withresonance modes of the second multimode resonator, where the secondmultimode resonator is a second left-handed transmission line. Also, themethod includes receiving pump signals, by the second multimoderesonator, wherein the pump signals are a first frequency sum through alast frequency sum, and generating, by the Josephson parametricconverter, a first qubit pair based on the first frequency sum through alast qubit pair based on the last frequency sum.

According to one embodiment, a method of generating a bell state usingphotons as quantum bits. The method includes providing a first multimoderesonator and a second multimode resonator both connected to adispersive nonlinear medium, where the first multimode resonator is afirst left-handed transmission line and the second multimode resonatoris a second left-handed transmission line, and where resonance modes areidentical in the first and second multimode resonators. The methodincludes receiving, by the second multimode resonator, a pump signal ata frequency sum, where the frequency sum is a summation of a resonancefrequency of the resonance modes plus another resonance frequency of theresonance modes. Also, the method includes generating a first photon anda second photon in an equal superposition of spatial states, where theequal superposition of the spatial states for the first and secondphotons are related to being in the first multimode resonator and thesecond multimode resonator.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a high-level schematic of a quantum microwave device accordingto an embodiment.

FIG. 2 is a circuit representation of a semi-infinite losslessleft-handed transmission line utilized in the multimode microwaveresonators of a multimode Josephson parametric converter according to anembodiment.

FIG. 3 is a schematic of the multimode Josephson parametric converteraccording to an embodiment.

FIG. 4 is a coplanar waveguide implementation of the multimode Josephsonparametric converter according to an embodiment.

FIG. 5 is a semi-coplanar stripline implementation of the multimodeJosephson parametric converter according to an embodiment.

FIG. 6 is a flow chart of a method of configuring a microwave apparatusaccording to an embodiment.

FIG. 7 is a flow chart of a method of operating the multimode Josephsonparametric converter as a multimode quantum limited amplifier accordingto an embodiment.

FIG. 8 is a schematic of a microwave quantum device for remoteentanglement of multiple qubits by measurement using the multimodeJosephson parametric converter according to an embodiment.

FIG. 9 is a flow chart of a method of remote entanglement of multiplequbits by measurement according to an embodiment.

FIG. 10 is a schematic of a microwave quantum device for remoteentanglement of multiple qubits by applying entangled photons to readoutthe state of the multiple qubits using the multimode Josephsonparametric converter according to an embodiment.

FIG. 11 is flow chart of a method of operating a multimode Josephsonparametric converter to generate multiple pairs of entangled photonsaccording to an embodiment.

FIG. 12 is a flow chart of a method of generating a bell state usingphotons as quantum bits according to an embodiment.

DETAILED DESCRIPTION

Embodiments disclose a quantum device based on the Josephson ringmodulator suitable for quantum information processing. The quantumdevice includes a Josephson ring modulator coupled to multimoderesonators implemented using metamaterial/left-handed transmissionlines, thereby forming a multimode Josephson parametric converter.

FIG. 1 is a high-level schematic of a quantum microwave device 100according to an embodiment. The quantum device 100 includes a multimodeJosephson ring modulator (JRM) 105 which is a nonlinear dispersiveelement based on Josephson tunnel junctions 102A, 102B, 102C, and 102Dthat can perform three-wave mixing of microwave signals at the quantumlimit. The JRM 105 consists of four nominally identical Josephsonjunctions 102A-102D arranged in a Wheatstone bridge configuration. Inorder to construct a non-degenerate parametric device that is themultimode Josephson parametric converter (JPC) 130, which is capable ofamplifying and/or mixing microwave signals at the quantum limit, the JRM105 is incorporated into two multimode microwave resonators at a radiofrequency (RF) current anti-node of the multiple of their eigenmodes.

One of the multimode microwave resonators is multimode resonator_a 115Aand the other is multimode resonator_b 115B. The multimode resonator_a115A is a left-handed transmission line with N unit cells, and themultimode resonator_b 115B is a left-handed transmission line with Munit cells as discussed further below. A coupling capacitor 110Aconnects the multimode resonator_a 115A to port_a 120A while thecoupling capacitor 110B connects the multimode resonator_b 115B toport_b 120B. The multimode JPC 130 includes both the multimoderesonator_a 115A and multimode resonator_b 115B, along with the JRM 105.

The performances (namely power gain G, dynamical bandwidth γ, andmaximum input power P_(max)) of the multimode JPC 130 are stronglydependent on the critical current I₀ of the Josephson junctions102A-102D of the JRM 105, the specific realization of theelectromagnetic environment (i.e., the microwave multimode resonator_a115A and microwave multimode resonator_b 115B), the coupling between theJRM 105 and the multimode resonators 115A and 115B, and the couplingbetween the multimode resonators to the feedlines.

The port_a 120A and/or port_b 120B may be microwave coaxial lines orwaveguides. Although not shown, other devices connected to the quantumdevice 100 may include hybrids, attenuators, circulators, isolators,lowpass microwave filters, bandpass microwave filters, infrared filters,and qubit-cavity systems.

FIG. 2 is a circuit of a semi-infinite lossless left-handed transmissionline which may be utilized in the construction of the multimodemicrowave resonator_a 115A and the multimode microwave resonator_b 115Baccording to an embodiment. The unit cell, e.g., unit cell 205A formicrowave multimode resonator_a 115A and unit cell 205B for microwavemultimode resonator_b 115B, includes a capacitor C₁ connected toinductor L_(l) where “1” represent left-handed transmission line. Theother end of the inductor L_(l) is connected to ground. The unit cell205A, 205B is connected to another unit cell, which is connected toanother unit cell, and so forth. The unit cell 205A is repeated N amountof times for the multimode resonator_a 115A, and the unit cell 205B isrepeated M amount of times for the multimode resonator_b 115B, as shownfurther below.

The dispersion relation of a left-handed transmission line reads

${\omega_{l}\left( k_{l} \right)} = \frac{1}{2\sqrt{L_{l}C_{l}}{\sin\left( \frac{k_{l}\Delta\; x}{2} \right)}}$

where Δx is the size of the unit cell, and k_(l) is the wave vector.

The phase and group velocity of the left-handed transmission line haveopposite orientation

${\frac{\partial{\omega_{l}(k)}}{\partial k} < 0},$where k is k_(l). One consequence of this relation is that inleft-handed transmission lines low-frequencies correspond to shortwavelengths. In contrast, in right-handed transmission lines where thedispersion relation increases with the wave vector, low-frequenciescorrespond to long wavelengths.

The characteristic impedance of the left-handed transmission line is

$Z_{l} = {\sqrt{\frac{L_{l}}{C_{l}}}.}$

Low-frequency bound of the left-handed transmission line is

$\omega_{IR} = {\frac{1}{2\sqrt{L_{l}C_{l}}}.}$

FIG. 3 is a schematic of the multimode Josephson parametric converter130 according to an embodiment. In FIG. 3, a 180° hybrid coupler 305Amay be connected to port_a 120A and a 180° hybrid coupler 305B may beconnected to port_b 120B.

A 180° hybrid is a 4-port microwave device which is reciprocal, matched,and ideally lossless. The 180° hybrid splits an input signal into twoequal amplitude outputs. When fed from its sum port (Σ) the 180° hybridprovides two equal-amplitude in-phase output signals, whereas when fedfrom its difference port (Δ), it provides two equal-amplitude 180°out-of-phase output signals.

One scenario assumes that there is a Signal (S) tone that lies withinthe bandwidth of one of the resonance modes of the multimode microwaveresonator_a which strongly couples to the JRM and is input through the Δport of the 180° hybrid 305A, and the 50 ohm (Ω) termination isconnected to the Σ port of the 180° hybrid 305A. It also assumes thatthere is an Idler (I) tone that lies within the bandwidth of one of theresonance modes of the multimode microwave resonator_b which stronglycouples to the JRM and is input through the Δ port of the 180° hybrid305B and a pump (P) tone input into the Σ port of the 180° hybrid 305B.Note that multiple pump tones at different frequencies may be utilizedin order to feed the device.

The two main operation modes of the device are amplification mode (withphoton gain) in which the applied pump frequency f_(P) satisfies therelationf _(P) =f _(l) +f _(S).

where f_(S) and f_(I) are the frequency of the Signal (S) and the Idler(I) tones respectively, and unitary frequency conversion mode (withoutphoton gain) in which the applied pump frequency f_(P) satisfies therelationf _(P) =|f _(I) −f _(S)|.

Different implementations of the quantum device with the multimodeJosephson parametric converter 130 are discussed herein according toembodiments.

In contrast, to dual-differential-mode (standard nondegenerate)state-of-the-art Josephson parametric converters made of right-handedtransmission lines, e.g., microstrip resonators, where the JRM isstrongly couples to the two fundamental resonance modes of the twophysical resonators of the device within the frequency band of interest,e.g. 5-15 GHz, the two multimode resonators of the multimode JPC 130,realized using metamaterial/left-handed transmission lines inembodiments (i.e., multimode resonator_a 115A and multimode resonator_b115B), can be designed and engineered such that the JRM 105 stronglycouples to multiple differential modes within the frequency band ofinterest. That is, each multimode resonator_a 115A and multimoderesonator_b 115B has multiple resonance modes within the frequency bandof interest, e.g. 5-15 GHz, many of which strongly couple to the JRM105, as opposed to the state-of-the-art JPCs having (only) twofundamental differential resonance modes for its resonators within theband of interest which strongly couple to the JRM.

Multimode means that the multimode resonator_a 115A has multipleresonance modes and that the multimode resonator_b 115B has multipleresonance modes within a certain frequency band of interest, e.g. 5-15GHz. This means that multimode resonator_a 115A is configured toresonate at multiple resonance frequencies from a first resonancefrequency through a last resonance frequency within a certain frequencyband of interest, which may include hundreds of resonance frequencies.Similarly, the multimode resonator_b 115B is configured to resonate atmultiple resonance frequencies from a first resonance frequency througha last resonance frequency within a certain frequency band of interest,e.g. 5-15 GHz, which may include hundreds of resonance frequencies.

One notable property of the left-handed transmission lines/resonators(respectively implemented as multimode resonator_a 115A and multimoderesonator_b 115B) is that they have a large density of modes (i.e.,density of resonance modes) close to their low-frequency bound ω_(IR)making them multimode resonators in the frequency band of interest. Forquantum measurements in superconducting devices, the band of interest isthe microwave band of approximately 5-15 gigahertz (GHz) (commonly usedfor qubit readout and measurement). The multimode resonator_a 115A andmultimode resonator_b 115B each can have a high density of resonancemodes (i.e., harmonics or resonance frequencies) between approximately5-15 GHz, which is beneficial for quantum measurements. In contrast, aright-handed transmission line (as a resonator) may have only oneharmonic (one frequency resonance mode) at about 10 GHz and the nextharmonic may be about 20 GHz (which is outside the 5-15 GHz microwaveband of interest) in the state-of-the-art. Frequency resonance modesoutside the 5-15 GHz microwave band of interest are not utilized tocarry quantum information (mainly because most superconducting qubitfrequencies fall within this range (i.e., fall within the band ofinterest), and many microwave generators, measuring devices, andmicrowave components are commercially available in this range), andtherefore, multimode resonator_a 115A and multimode resonator_b 115B inembodiments may each have several tens or hundreds of frequencyresonance modes (i.e., high density of modes) between 5-15 GHz which canbe utilized to process quantum information using the multimode JPC 130.

In general, the density of modes of left-handed transmission lineresonators at a given angular resonance frequency ω is proportional tothe number of units cells in the resonator and inversely proportional tothe low-frequency bound ω_(IR).

It is to be noted that not all of the multiple resonance modes ofmultimode resonator_a and multimode resonator_b modes that fall within acertain band of interest, e.g., 5-15 GHz, strongly couple to the JRM atthe center, i.e., have an RF-current antinode at the location of theJRM. Hence, the resonance modes which strongly couple to the JRM are asubset (approximately half) of the available resonance modes within theband of interest. Consequently, not all of the resonance modes ofmultimode resonator_a and multimode resonator_b that fall within theband of interest can be utilized in order to perform three-wave mixing,which forms the basis for the various quantum information processingoperations enabled by this multimode device. In other words, the termmultiple modes of multimode resonator_a and multimode resonator_b usedin this disclosure mainly refers to those which strongly couple to JRMwithin the band of interest.

In one implementation, the multimode resonator_a 115A and multimoderesonator_b 115B may each have between 5 to 20 frequency resonance modesin the range 5-10 GHz which strongly couple to the JRM. In anotherimplementation, the multimode resonator_a 115A and multimode resonator_b115B may each have between 20-50 frequency resonance modes in the range5-10 GHz which strongly couple to the JRM. In yet anotherimplementation, the multimode resonator_a 115A and multimode resonator_b115B may each have 50-100 frequency resonance modes in the range 5-10GHz which strongly couple to the JRM.

Since the multimode resonator_a 115A and multimode resonator_b 115B mayeach have multiple resonance modes which strongly couple to the JRM inthe range 5-10 GHz (e.g. 5-100 frequency resonance modes), this allowsthe multimode JPC 130 to be useful in various intriguing applications inthe area of quantum information processing beyond the capabilities ofstandard dual-differential mode JPCs, such as generation of remoteentanglement between multiple qubits, generation of multiple pairs ofentangled photons, amplification of multiple microwave signals at thequantum limit, and performing unitary frequency conversion betweenmultiple propagating microwave signals at different frequencies.

FIG. 4 is an exemplary coplanar waveguide implementation of themultimode Josephson parametric converter 130 according to an embodiment.

The multimode JPC 130 includes multimode resonator_a 115A (left-handedtransmission line) comprising lumped-element inductors L_(a) (as theinductor L_(l)) and lumped-element capacitors C_(a) (as the capacitorsC_(l)). Similarly, the multimode JPC 130 includes multimode resonator_b115B (left-handed transmission line) comprising lumped-element inductorsL_(b) (as the inductor L_(l)) and lumped-element capacitors C_(b) (asthe capacitors C_(l)).

The multimode resonator_a 115A (left-handed transmission line) isconnected to the left and right nodes of the Josephson ring modulator150. The multimode resonator_a 115A connects to port_a 120A. In themultimode resonator_a 115A, the unit cell 205A includes two inductorsL_(a) connected to the capacitor C_(a). One end of the two inductorsL_(a) is connected to each other and the capacitor C_(a), while theother end of the inductors L_(a) is connected to the ground plane 405.This configuration of the unit cell 205A repeats N amount of times inthe multimode resonator_a 115A as shown in FIG. 4. It should be notedthat the use of two inductors in each unit cell is mainly for thepurpose of keeping the device symmetric with regard to connection toground. However, the use of one inductor connected to ground is alsocontemplated in one implementation.

The multimode resonator_b 115B (left-handed transmission line) isconnected to the top and bottom nodes of the Josephson ring modulator150. The multimode resonator_b 115B connects to port_b 120B. In themultimode resonator_b 115B, the unit cell 205B includes two inductorsL_(b) connected to the capacitor C_(b). One end of the two inductorsL_(b) is connected to each other and the capacitor C_(b), while theother end of the inductors L_(b) is connected to the ground plane 405.In the multimode resonator_b 115B, this configuration of the unit cell205B repeats M amount of times in the multimode resonator_b 115B asshown in FIG. 4. It should be noted that the use of two inductors ineach unit cell is mainly for the purpose of keeping the device symmetricwith regard to connection to ground. However, the use of one inductorconnected to ground is also contemplated in an implementation.

As discussed in FIG. 2, the ports_a 115A and ports_b 115B may be fedusing 180° hybrids 305A and 305B (not shown in FIG. 4). The ports_a 115Aand ports_b 115B may be coaxial cables or coplanar waveguides or may bemicrostrips or striplines with a center conductor and outside conductorseparated by a dielectric material. For ports_a 115A, the centerconductor connects to the left and right sides of the multimoderesonator_a 115A through coupling capacitors 110A, while the outsideconductor is connected to the ground plane 405. For ports_b 115B, thecenter conductor connects to the top and bottom sides of the multimoderesonator_b 115B through coupling capacitors 110B, while the outsideconductor is connected to the ground plane 405.

FIG. 5 is an exemplary semi-coplanar stripline implementation of themultimode Josephson parametric converter 130 according to an embodiment.

The multimode JPC 130 includes multimode resonator_a 115A (left-handedtransmission line) comprising inductor L_(a) (as the inductor L_(l)) andcapacitors C_(a) (as the capacitors C_(l)). Similarly, the multimode JPC130 includes multimode resonator_b 115B (left-handed transmission line)comprising inductors L_(b) (as the inductor L_(l)) and capacitors C_(b)(as the capacitors C_(l)).

The lumped-element side of the multimode resonator_a 115A is connectedto the left node of the Josephson ring modulator 150, while the rightnode is connected to the conducting plane 406. The lumped-element sideof the multimode resonator_a 115A and the conducting plane 406 connectto the 180 hybrid 305A. In the multimode resonator_a 115A, the unit cell205A includes inductor L_(a) connected to the capacitor C_(a). One endof the inductor L_(a) is connected to the capacitor C_(a), while theother end of the inductor L_(a) is connected to the conducting plane406. This configuration of the unit cell 205A repeats N amount of timesin the multimode resonator_a 115A as shown in FIG. 5. Although FIG. 5illustrates the left node connected to the lumped-element side of themultimode resonator_a 115A and the right node connected to theconducting plane 406, this configuration can be interchanged such thatthe lumped-element side of the multimode resonator_a 115A is connectedto the right node and the conducting plane 406 is connected to the leftnode.

The lumped-element side of the multimode resonator_b 115B is connectedto the top node of the Josephson ring modulator 150, while the bottomnode is connected to the conducting plane 407. The lumped-element sideof the multimode resonator_b 115B and conducting plane 407 connect tothe port_b 115B. In the multimode resonator_b 115B, the unit cell 205Bincludes inductor L_(b) connected to the capacitor C_(b). One end of theinductor L_(b) is connected to the capacitor C_(b), while the other endof the inductor L_(b) is connected to the conducting plane 407. Thisconfiguration of the unit cell 205B repeats M amount of times in themultimode resonator_b 115B as shown in FIG. 5. Although FIG. 5illustrates the top node connected to the lumped-element side of themultimode resonator_b 115B and the bottom node connected to theconducting plane 407, this configuration can be interchanged such thatthe lumped-element side of the multimode resonator_b 115B is connectedto the top node and the conducting plane 407 is connected to the bottomnode.

FIG. 6 is a method of configuring a microwave apparatus (such asmultimode JPC 130) according to an embodiment. Reference can be made toFIGS. 1-5.

At block 605, a first multimode resonator (i.e., multimode resonator_a115A) is connected to a Josephson ring modulator 150, where the firstmultimode resonator is made of a first left-handed transmission line.

At block 610, a second multimode resonator (i.e., multimode resonator_b115B) is connected to the Josephson ring modulator 150, where the secondmultimode resonator is made of a second left-handed transmission line.

The first multimode resonator (i.e., multimode resonator_a 115A)comprises a plurality of first resonance modes which strongly couple tothe JRM within a certain frequency band of interest, and the secondmultimode resonator (i.e., multimode resonator_b 115B) comprises aplurality of second resonance modes which strongly couple to the JRMwithin the same frequency band of interest.

A number of the plurality of first resonance modes in the firstmultimode resonator which strongly couple to the JRM within a certainfrequency band of interest is equal to a number of a plurality of secondresonance modes in the second multimode resonator which strongly coupleto the JRM within the same frequency band of interest. For example, thenumber of frequency resonance modes which strongly couple to the JRMwithin a certain frequency band of interest is equal in multimoderesonator_a 115A and multimode resonator_b 115B.

The number of the plurality of first resonance modes in the firstmultimode resonator which strongly couple to the JRM within a certainfrequency band of interest does not equal to the number of the pluralityof second resonance modes in the second multimode resonator whichstrongly couple to the JRM within the same frequency band of interest.For example, the multimode resonator_a 115A or the multimode resonator_b115B may have within a certain frequency band of interest more frequencyresonance modes which strongly couple to the JRM than the other.

The first multimode resonator_a 115A comprises N amount of first unitcells 205A, and the second multimode resonator_b 115B comprises M amountof second unit cells 205B. Neither N nor M is equal to zero. In oneimplementation, N equals M, and in another implementation N does notequal M.

Each of the first unit cells 205A and each of the second unit cells 205Brespectively comprises a capacitor (C_(a), C_(b)) connected to one endof an inductor (L_(a), L_(b)), while another end of the inductor (L_(a),L_(b)) is connected to ground 405 or conducting planes 406, 407, asdepicted is FIGS. 2, 4 and 5.

The Josephson ring modulator 150 comprises a first pair of nodes (e.g.,left and right nodes JRM 150) opposite one another and a second pair ofnodes (e.g., top and bottom nodes of the JRM 150) opposite one another.The first multimode resonator_a 115A is connected to the first pair ofnodes. One of the first pair of nodes is connected to the lumped-elementside of the resonator, and the conducting plane 406 is connected toanother one of the first pair of nodes, as depicted in FIG. 5. Thesecond multimode resonator_b 115B is connected to the second pair ofnodes. One of the second pair of nodes is connected to thelumped-element side of the resonator and the conducting plane 407 isconnected to another one of the second pair of nodes, as depicted inFIG. 5.

Each of the first unit cells 205A and each of the second unit cells 205Brespectively comprises a first inductor (first L_(a), first L_(b)), asecond inductor (second L_(a), second L_(b)), and a capacitor (C_(a),C_(b)), as depicted in FIG. 4. The first ends of the first inductor andthe second inductor are connected together, while second ends of thefirst inductor and the second inductor are connected to ground, and thecapacitor is connected to the first ends, as depicted in FIG. 4.

The Josephson ring modulator 150 comprises a first pair of nodesopposite one another and a second pair of nodes opposite one another ina Wheatstone bridge. The first multimode resonator_a 115A is connectedto the first pair of nodes, and the second multimode resonator_b 115B isconnected to the second pair of nodes.

The first unit cells are connected to one another in series, and thesecond unit cells are connected to one another in series.

In one implementation, a capacitance and an inductance in each of thefirst unit cells 205 (in the multimode resonator_a 115A) are differentfrom a capacitance and an inductance in each of the second unit cells(in the multimode resonator_b 115B). Since the unit cells 205A in themultimode resonator_a 115A are different from the unit cells 205B in themultimode resonator_b 115B, the multimode resonator_a 115A has differentresonance modes and resonance frequencies than the multimode resonator_b115B.

In another implementation, the capacitance and the inductance in each ofthe first unit cells (in the multimode resonator_a 115A) match thecapacitance and the inductance in each of the second unit cells (in themultimode resonator_b 115B).

The lumped-element inductances and capacitances used in each multimoderesonator can vary from one unit cell to another. Such perturbation tothe periodic structure of the multimode resonator can be used in orderto alter the frequency spacing between certain eigenmodes of themultimode resonator.

The lumped-element inductances used in the design of the left-handedtransmission lines of the multimode resonators, e.g. L_(a) and L_(b),can be implemented using narrow superconducting wires in a meanderconfiguration. The total inductance of the superconducting wire may be acombination of geometric and kinetic inductances. The lumped-elementinductances used in the design of the left-handed transmission lines ofthe multimode resonators can also be implemented as an array of largeJosephson Junctions.

The lumped-element capacitances used in the design of the left-handedtransmission lines of the multimode resonators, e.g. C_(a) and C_(b),can be implemented as interdigitated capacitors or plate capacitors witha dielectric layer deposited between two electrodes along the centerconductor of the left-handed transmission line.

According to an embodiment, the multimode Josephson parametric converter130 is a multimode quantum-limited amplifier that operates in themicrowave domain and can be used in quantum information processing andquantum computing. The multimode JPC 130 is a device that can amplifymultiple signals in parallel. The multimode JPC 130 can also be utilizedto remotely entangle multiple superconducting qubits. Additionally,multimode JPC 130 has benefits in scalable quantum computingarchitectures over other alternatives. The multimode JPC 130 can alsoplay a major role in quantum computation schemes that use microwavephotons as qubits.

There are not any state-of-the-art devices that have similarcapabilities while supporting multiple microwave eigenmodes within thefrequency range of interest between 5-15 GHz. A state-of-the-artJosephson parametric converter can support only two eigenmodes withinthe same frequency range.

Various beneficial applications of utilizing the multimode Josephsonparametric converter 130 are discussed below. For ease of understanding,sub-titles or sub-headings are provided below. The sub-titles are forexplanation purposes and not limitation.

Multimode Quantum Limited Amplifier

The multimode Josephson parametric converter 130 is configured tooperate as a multimode quantum limited amplifier. The multimodeJosephson parametric converter 130 is configured to amplify multiplequantum signals at different frequencies in parallel (i.e.,simultaneously) at the quantum limit. For example, the multiple quantumsignals may be multiple Signal (S) quantum signals (corresponding toresonance frequencies on multimode resonator_a 115A) and Idler (I)quantum signals (corresponding to resonance frequencies on multimoderesonator_b 115B) input respectively into the multimode resonator_a 115Avia port_a 120A and into multimode resonator_b 120B via port_b 115B.

The multimode Josephson parametric converter 130 is configured to selectwhich input signals (i.e., input quantum signals) are amplified inreflection and which are reflected off the JPC 130 without gain.

As an example, consider a multimode JPC 130 with multimode resonator_a115A and multimode resonator_b 115B having resonance frequencies f₁^(a), f₂ ^(a), f₃ ^(a), f_(r) ^(a), . . . and f₁ ^(b), f₂ ^(b), f₃ ^(b),f₄ ^(b), . . . which couple to the JRM 105 respectively. Bysimultaneously pumping the multimode JPC 130 (device) with three pumptones whose frequencies f₁ ^(p), f₂ ^(p), f₄ ^(p) satisfy the relationsf₁ ^(p)=f₁ ^(a)+f₁ ^(b), f₂ ^(p)=f₂ ^(a)+f₄ ^(p)=f₄ ^(a)+f₄ ^(b), theinput quantum signals on port_a 120A and port_b 120B at frequencies f₁^(a), f₂ ^(a), f₄ ^(a) and f₁ ^(b), f₂ ^(b), f₄ ^(b) respectively areamplified (by gain G) in reflection, whereas input signals atfrequencies f₃ ^(a), f₃ ^(b) are reflected off the JPC 130 without gainG. Amplified in reflection means that the respective quantum signals atfrequencies f₁ ^(a), f₂ ^(a), f₄ ^(a) and frequencies f₁ ^(b), f₂ ^(b),f₄ ^(b) are amplified back through their respective port_a 120A andport_b 120B in which the multiple quantum signals were initially input.

According to an implementation, in order for the multimode JPC 130device to operate properly as an amplifier (i.e., avoid pump depletioneffects, such as decrease in the dynamic range), the pump tones are tosatisfy the stiff pump approximation, where the applied pump tones arenonresonant with the resonance modes of the JPC 130. In other words, thepump frequencies f₁ ^(p), f₂ ^(p), f₃ ^(p), . . . should not coincidewith the frequency resonance modes of multimode resonator_a 115A andmultimode resonator_b 115B.

According to one implementation, in order to prevent frequencycollisions between the frequency of the pump (P) tones that need to beapplied in order to amplify certain resonance frequencies (of themultimode resonators 115A and 115B) and the high resonance frequenciesof the multimode resonators 115A and 115B, certain mode engineeringtechniques can be applied in the design of the multimode resonators 115Aand 115B. One example of such a mode engineering technique is to reducethe number of unit cells 205A and 205B in the multimode resonators 115Aand 115B, respectively, in order to reduce the density of modes in thefrequency band at which the pump frequencies need to be applied. It isnoted that this measure can also shift the resonance frequencies whichthe operators are interested in amplifying, and as a result, shift thepump frequencies as well. Another example is to perturb (i.e., change)the periodic structure of the multimode resonators 115A and 115B bymodifying the inductance (e.g., of inductors L_(a), L_(b)) and/or thecapacitance (of capacitors C_(a), C_(b)) of certain unit cells 205A,205B.

According to one implementation, it is further noted that not all of thefrequency resonance modes of multimode resonators 115A and 115B areexpected to have an RF-current antinode at the JRM 105 position that isrequired for amplification. Therefore, the frequency resonance modesthat can be amplified may (only) be a subgroup of all the frequencyresonance modes of the multimode JPC 130 (device) in one implementation.

FIG. 7 is a flow chart 700 of a method of operating the multimodeJosephson parametric converter 130 as a multimode quantum limitedamplifier according to an embodiment.

At block 705, the multimode resonator_a 115A and the multimoderesonator_b 115B (in the multimode Josephson parametric converter 130)receive multiple quantum signals in parallel which lie within thebandwidths of different resonance modes via port_a 120A and port_b 120B.

At block 710, the multimode Josephson parametric converter 130 amplifiessimultaneously the multiple quantum signals, according to pump drives(tones) applied to, e.g., port_a 120A connected to multimode resonator_a115A of the multimode Josephson parametric converter 130.

At block 715, the multimode parametric converter 130 is configured toreflect the amplified multiple quantum signals which lie within thebandwidths of the different resonance frequencies, according to theapplied pump drives. The amplified quantum signals are reflected back tothe ports 120A and 120B on which they were input.

A first group of the different resonance modes of the first multimoderesonator_a 115A configured to resonate at the first group of thedifferent resonance frequencies (i.e., the first group of the differentresonance frequencies correspond to the resonance modes for themultimode resonator_a 115A). The first multimode resonator_a 115A is afirst left-handed transmission line.

A second group of the different resonance modes of the second multimoderesonator_b 115B configured to resonate at the second group of thedifferent resonance frequencies (i.e., the second group of the differentresonance frequencies correspond to the resonance modes for themultimode resonator_b 115B). The second multimode resonator_b 115 is asecond left-handed transmission line.

Each of the pump drives (input into the JPC 130) is a frequency sum ofone of the different resonance frequencies in the first group and one ofthe different resonance frequencies in the second group which couple tothe JRM 105, such that the multiple quantum signals at the one of thefirst group and the one of the second group are amplified.

A first pump signal is a first frequency sum (e.g., f₁ ^(p)=f₁ ^(a)+f₁^(b)) of a first resonance frequency in the first group (e.g., f₁ ^(a))plus a first resonance frequency in the second group (e.g., f₁ ^(b)). Asecond pump signal is a second frequency sum of a second resonancefrequency in the first group plus a second resonance frequency in thesecond group. A last pump signal is a last frequency sum of a lastresonance frequency in the first group plus a last resonance frequencyin the second group. It is noted that there are many more combinationsof frequencies that can be used as the frequency sum, and the examplefrequency sum (e.g., f₁ ^(p)=f₁ ^(a)+f₁ ^(b)) is provided forexplanation purposes. For example, another frequency sum may be f₁₂^(p)=f₁ ^(a)+f₂ ^(b), where the first resonance frequency in the firstgroup may be f₁ ^(a) and the second resonance frequency in the secondgroup may be f₂ ^(b).

At least one of the first through last resonance frequencies in thefirst group and at least one of the first through last resonancefrequencies in the second group is the same. At least one of the firstthrough last resonance frequencies in the first group and the firstthrough last resonance frequencies in the second group is different.

The multiple quantum signals at the different frequencies range fromabout 5-15 GHz. The multimode Josephson parametric converter isconfigured to amplify simultaneously (or nearly simultaneously) themultiple quantum signals which lie within the bandwidths of thedifferent resonance modes via the first multimode resonator_a 115A andthe second multimode resonator_b 115B, where the first and secondmultimode resonators 115A, 115B are coupled to a dispersive nonlinearmedium 105.

Remote Entanglement of Multiple Qubits by Measurement Scheme

FIG. 8 is a schematic of a microwave quantum device 800 for remoteentanglement of multiple qubits by measurement using the multimode JPC130 according to an embodiment. For the sake of simplicity, the detailsof the multimode JPC 130 are not shown in FIG. 8. For details of themultimode JPC 130, reference can be made to FIGS. 1-6 discussed herein.

In FIG. 8, the multimode JPC 130 has port_a 115A connected to acirculator 815A and port_b 115B connected to circulator 815B. The pumptones/drives are multiple pump frequencies input into the multimode JPC130 as discussed further below. Output field quadratures I_(a)(t) and/orQ_(a)(t) are measured via mixer 835A, high electron mobility transistor(HEMT) 830A, the circulator 815A (assumed to be lossless), and multimodeJPC 130. Similarly, output field quadratures I_(b)(t), and/or Q_(b)(t)are measured via mixer 835B, high electron mobility transistor (HEMT)830B, the circulator 815B (assumed to be lossless), and multimode JPC130.

It is assumed that the resonance frequencies (f₁ ^(a), f₂ ^(a), f₃ ^(a),. . . , f_(n) ^(a)) of the multimode resonator_a 115A of port_a 120Athat couple to the JRM 105 (within the JPC 130) coincide with (i.e., arethe same as) the readout frequencies of the readout resonators802_1-802_n coupled to bus_a 810A. The readout resonators 802_1-802_nmay be capacitively coupled to the bus_a 810A.

It is assumed that the resonance frequencies (f₁ ^(b), f₂ ^(b), f₃ ^(b),. . . , f_(n) ^(b)) of the multimode resonator_b 115B of port_b 120Bthat couple to the JRM 105 coincide with the readout frequencies of thereadout resonators 804_1-804_n coupled to bus_b 810B. The readoutresonators 804_1-804_n may be capacitively coupled to the bus_a 810B.

The qubits 820A include q₁ ^(a), q₂ ^(a), q₃ ^(a), . . . , q_(n) ^(a),each capacitively coupled to its own readout resonator 802_1, 802_2,802_3, . . . 802_n. Similarly, the qubits 820B include q₁ ^(b), q₂ ^(b),q₃ ^(b), . . . q_(n) ^(b), each capacitively coupled to its own readoutresonator 804_1, 804_2, 804_3, . . . 804_n. In FIG. 8, applying thefrequency f₁ ^(a) at input 1 (IN1) has the dual effect of causingreadout resonator 802_1 to read out its qubit q₁ ^(a) and causing themultimode resonator_a 115A to resonate at its resonance frequency f₁^(a). Analogously, applying the frequencies (f₁ ^(a), f₂ ^(a), f₃ ^(a),. . . , f_(n) ^(a)) in parallel at input 1 (IN1) has the dual effect ofcausing readout resonators 802_1, 802_2, 802_3, . . . , 802_n to readout their capacitively coupled qubit q₁ ^(a), q₂ ^(a), q₃ ^(a), . . .q_(n) ^(a) respectively and causing the multimode resonator_a 115A toresonate at its resonance frequencies (f₁ ^(a), f₂ ^(a), f₃ ^(a), . . ., f_(n) ^(a)). Each readout resonator 802 respectively shares the sameresonance frequency as one of the resonance modes in the multimoderesonator_a 115A, such that a signal with this same frequency causesboth to resonate.

Similarly, applying the frequency f₁ ^(b) at input 2 (IN2) has the dualeffect of causing readout resonator 804_1 to read out its qubit q₁ ^(b)and causing the multimode resonator_b 115B to resonate at its resonancefrequency f₁ ^(b). Analogously, applying the frequencies (f₁ ^(b), f₂^(b), f₃ ^(b), . . . , f_(n) ^(b)) in parallel at input 2 (IN2) has thedual effect of causing readout resonators 804_1, 804_2, 804_3, . . . ,804_n to read out their capacitively coupled qubit q₁ ^(b), q₂ ^(b), q₃^(b), . . . q_(n) ^(b) respectively and causing the multimoderesonator_b 115B to resonate at its resonance frequencies (f₁ ^(b), f₂^(b), f₃ ^(b), . . . , f_(n) ^(b)). Each readout resonator 804respectively shares the same resonance frequency as one of the resonancemodes in the multimode resonator_b 115B, such that a signal with thissame frequency causes both to resonate.

The multimode JPC 130 is configured to remotely entangle pairs of qubits(individually from qubits 820A and 820B) coupled to buses 810A and 810Bvia measurement, by applying multiple pump tones to the multimode JPC130 where the pump tone frequencies (f₁ ^(p), f₂ ^(b), f₃ ^(p), . . . ,f_(m) ^(p)) correspond to the frequency sum of the readout frequenciesof the pairs of qubits in qubits 820A and 820B that are to be entangled.With respect to applying the frequencies (f₁ ^(a), f₂ ^(a), f₃ ^(a), . .. , f_(n) ^(a)) and (f₁ ^(b), f₂ ^(b), f₃ ^(b), . . . , f_(n) ^(b)) andapplying the pump tone frequencies (f₁ ^(p), f₂ ^(p), f₃ ^(p), . . . ,f_(m) ^(p)), it should be appreciated that the index m is larger thanthe index n, because there are more combinations for the pump tonefrequencies than individual resonance frequencies. For explanationpurposes, a few examples are provided for illustration and notlimitation.

Example 1

In this case, the microwave quantum device 800 is entangling the qubitpairs (q_(i) ^(a), q_(i) ^(b)) where i ∈{1, 2, . . . n} and f_(i)^(a)≠f_(i) ^(b). Accordingly, the applied pump frequencies are given byf_(i) ^(p)=f_(i) ^(a)+f_(i) ^(b). In this example, each of the qubits820A is entangled respectively with its corresponding qubit in qubits820B, such that, e.g., there are entangled qubit pairs (q₁ ^(a), q₁^(b)), (q₂ ^(a), q₂ ^(b)), (q₃ ^(a), q₃ ^(b)) through entangled qubitpair (q_(n) ^(a), q_(n) ^(b)).

Example 2

In this case, microwave quantum device 800 is entangling the qubit pairs(q_(i) ^(a), q_(j) ^(b)) where i, j ∈ {1, 2, . . . n} and f_(i)^(a)≠f_(j) ^(b) given by f_(ij) ^(p)=f_(i) ^(a)+f_(j) ^(b).

It is noted that entanglement by the microwave quantum device 800 is notlimited to pairs of qubits. For example, the microwave quantum device800 is configured to entangle three qubits q₁ ^(a), q₂ ^(b), q₃ ^(b)coupled to readout resonators 802_1, 804_2, 804_3 respectively withfrequencies f₁ ^(a), f₂ ^(b), f₃ ^(b) by applying two pump tones atfrequencies f₁₂ ^(p)=f₁ ^(a)+f₂ ^(b) and f₁₃ ^(p)=f₁ ^(a)+f₃ ^(b)sequentially. Thus, using this method of entanglement, the microwavequantum device 800 is configured to remotely entangle n different qubitsby applying sequentially n−1 pumps that satisfy the sum condition of thereadout frequencies of those qubits.

FIG. 9 is a flow chart 900 of a method of remote entanglement ofmultiple qubits by measurement according to an embodiment.

At block 905, the first multimode resonator_a 115A in the multimodeJosephson parametric converter 130 is configured to receive a firstgroup of readout signals at different resonance frequencies (f₁ ^(a), f₂^(a), f₃ ^(a), . . . , f_(n) ^(a)) or lie within the bandwidths ofresonance modes of the first multimode resonator_a 115A, where the firstmultimode resonator is a first left-handed transmission line.

At block 910, the second multimode resonator_b 115B in the multimodeJosephson parametric converter 130 is configured to receive a secondgroup of readout signals at different resonance frequencies (f₁ ^(b), f₂^(b), f₃ ^(b), . . . , f_(n) ^(b)) or lie within the bandwidths ofresonance modes of the second multimode resonator, where the secondmultimode resonator is a second left-handed transmission line.

At block 915, the second multimode resonator_b 115B is configured toreceive pump signals (f₁ ^(p), f₂ ^(p), f₃ ^(p), . . . , f_(n) ^(p)),where the pump drives are a first (pump) frequency sum (e.g., f₁ ^(p)=f₁^(a)+f₁ ^(b)), a second (pump) frequency sum (e.g., f₂ ^(p)=f₂ ^(a)+f₂^(b)), through a last (pump) frequency sum (e.g., f_(n) ^(p)=f_(n)^(a)+f_(n) ^(b)). It is noted that n denotes the last number of theparticular series of items. As understood by one skilled in the art,there are many other combinations that may be utilized and the examplefrequency sums are only for explanation purposes. For example, anotherfrequency sum may be f₂₃ ^(p)=f₂ ^(a)+f₃ ^(b), f_(ln) ^(p)=f₁ ^(a)+f_(n)^(b), and so forth.

At block 920, the Josephson parametric converter 130 is configured togenerate a first qubit pair (e.g., (q₁ ^(a), q₁ ^(b)) based on the firstfrequency sum, a second qubit pair (e.g., (q₂ ^(a), q₂ ^(b)) based onthe second frequency sum, through a last qubit pair (e.g., (q_(n) ^(a),q_(n) ^(b))) based on the last frequency sum. As understood by oneskilled in the art, these example qubit pairs are illustrated only forexplanation purposes, and many other combinations of qubit pairs can bematched. For example, other qubit pairs may include (q₁ ^(a), q_(n)^(b)), (q_(n) ^(a), q₂ ^(b)), and so forth.

The first frequency sum is a summation of one resonance frequency of thefirst group plus one resonance frequency of the second group, where thesecond frequency sum is a summation of another resonance frequency ofthe first group plus another resonance frequency of the second group,and where the last frequency sum is a summation of yet another resonancefrequency of the first group plus yet another resonance frequency of thesecond group. It is understood that there are numerous combinations, andembodiments are not meant to be limited.

The first group of readout signals is received from first readoutresonators 802, and the first readout resonators 802 are coupled tofirst qubits 820A. The second group of readout signals is received fromsecond readout resonators 804, and the second readout resonators 804 arecoupled to second qubits 820B.

One qubit (e.g., q₁ ^(a)) in the first qubits 820A has been readout atthe one resonance frequency of the first group and one qubit (e.g., q₁^(b)) in the second qubits 820B has been readout at the one resonancefrequency of the second group, such that the first qubit pair (e.g., (q₁^(a), q₁ ^(b))) is the one qubit in the first qubits and the one qubitin the second qubits, in response to the first frequency sum of the pumpsignals. Another qubit (e.g., q₂ ^(a)) in the first qubits 820A has beenreadout at the another resonance frequency of the first group andanother qubit (e.g., q₂ ^(b)) in the second qubits 820B has been readoutat the another resonance frequency of the second group, such that thesecond qubit pair (e.g., (q₂ ^(a), q₂ ^(b))) is the another qubit in thefirst qubits 820A and the another qubit in the second qubits 820B, inresponse to the second frequency sum of the pump signals. Also, yetanother qubit (e.g., q_(n) ^(a)) in the first qubits 820A has beenreadout at the yet another resonance frequency of the first group andyet another qubit (e.g., q_(n) ^(b)) in the second qubits 820B has beenreadout at the yet another resonance frequency of the second group, suchthat the third qubit pair (e.g., (q_(n) ^(a), q_(n) ^(b))) is the yetanother qubit in the first qubits and the yet another qubit in thesecond qubits, in response to the last frequency sum of the pumpsignals. As noted herein, combinations are pointed out for ease ofunderstanding, but embodiments are not limited to the examplecombination. It is understood that there are many more combinations.Moreover, there may be every pair combination taken from resonator_a115A and resonator_b 115B, where the pump frequency (i.e., frequencysum) needs to have two indices (one for the qubit on bus_a 810A and onefor the qubit on bus_b 810B).

The first readout resonators 802_1 through 802_n have readout resonatorfrequencies (f₁ ^(a), f₂ ^(a), f₃ ^(a), . . . , f_(n) ^(a)) thatcoincide with the resonance frequencies of the first multimoderesonator_1 115A. The second readout resonators 804_1 through 804_n havereadout resonator frequencies (f₁ ^(b), f₂ ^(b), f₃ ^(b), . . . , f_(n)^(b)) that coincide with the resonance frequencies of the secondmultimode resonator_b 115B.

Both the resonance modes of the first multimode resonator and thereadout resonator frequencies of the first readout resonator range fromabout 5-15 GHz. Both the resonance modes of the second multimoderesonator and the readout resonator frequencies of the second readoutresonator range from about 5-15 GHz.

In the section of remote entanglement of multiple qubits by measurementscheme, the qubits 820A, 820B are read out first and then the JPC 130entangles pairs of qubits coupled to readout resonators and buseslocated on either side (port) of the multimode JPC 130.

Remote Entanglement Scheme of Multiple Qubits by Applying EntangledPhotons to Readout the State of the Multiple Qubits

The sections (for the generation of multiple pairs of entangled photonsand the remote entanglement of multiple qubits by applying entangledphotons) generate entangled photons first, such that the entangledphotons can be further utilized, such as to readout out qubits. Theorder of these sections (the generation of multiple pairs of entangledphotons and the remote entanglement of multiple qubits by applyingentangled photons) is reversed compared to the section of remoteentanglement of multiple qubits by measurement scheme.

FIG. 10 is a schematic of a microwave quantum device 1000 for remoteentanglement of multiple qubits by applying entangled photons to readoutthe state of the multiple qubits using the multimode JPC 130 accordingto an embodiment. For the sake of simplicity, the details of themultimode JPC 130 are not shown in FIG. 10. For details of the multimodeJPC 130, reference can be made to FIGS. 1-6 discussed herein. Also,certain components of microwave device 800 are included in the microwavedevice 1000. However, the ports of the circulators are different as wellas the rotation.

In FIG. 10, input IN1 connects directly to circulator 1015A, such thatthe circulator 1015A first transmits the input signal of input IN1directly to port_a 120A. Similarly, input IN2 connects directly tocirculator 1015B, such that the circulator 1015B first transmits theinput signal of input IN2 directly to port_b 120B.

The reflected output of port_a 120A is transmitted to the bus_a 810A viathe circulator 1015A and then to the measurement equipment for measuringthe output field quadratures Ia(t) and/or Qa(t). Similarly, thereflected output of port_b 120B is transmitted to the bus_b 810B via thecirculator 1015B and then to the measurement equipment for measuring theoutput field quadratures Ib(t) and/or Qb(t).

In the microwave quantum device 1000, the resonance frequencies of themultimode resonator_a 115A of (connected to) port_a 120A that couple tothe JRM 105 coincide with the readout frequencies of the readoutresonators 802 coupled to bus_a 810A, i.e., f₁ ^(a), f₂ ^(a), f₃ ^(a), .. . , f_(n) ^(a). Similarly, the resonance frequencies of theresonator_b 115B of (connected to) port_b 120B that couple to the JRM105 coincide with the readout frequencies of the readout resonators 804coupled to bus_b 810B, i.e., f₁ ^(b), f₂ ^(b), f₃ ^(b), . . . , f_(n)^(b). Accordingly, the microwave quantum device 1000 is configured toremotely entangle pairs of qubits coupled to bus_a 810A and bus_b 810Bby applying entangled pairs of photons generated by the JPC 130 at thecorresponding readout frequencies. This can be achieved by applyingmultiple pump tones to the multimode JPC130 whose frequencies f₁ ^(p),f₂ ^(p), f₃ ^(p), . . . , f_(m) ^(p) correspond to the frequency sum ofthe readout frequencies of the pairs of qubits that are to be entangled.

Example 1

In order for the microwave quantum device 1000 to entangle the qubitpairs (q_(i) ^(a), q_(i) ^(b)) where i ∈ {1, 2, . . . n} and f_(i)^(a)≠f_(i) ^(b), the microwave quantum device 1000 receives the appliedpump drives whose frequencies are given by f_(i) ^(p)=f_(i) ^(a)+f_(i)^(b).

Example 2

In order for the microwave quantum device 1000 to entangle the qubitpairs (q_(i) ^(a), q_(j) ^(b))) where i, j ∈ {1, 2, . . . n} and f_(i)^(a)≠f_(j) ^(b), the microwave quantum device 1000 receives the appliedpump drives whose frequencies are given by f_(ij) ^(p)=f_(i) ^(a)+f_(j)^(b).

It is noted that entanglement via microwave quantum device 1000 is notlimited to pairs of qubits. For example, the microwave quantum device1000 is configured to entangle three qubits q₁ ^(a), q₂ ^(b), q₃ ^(b)coupled to readout resonators 802_1, 804_2, 804_3 with frequencies f₁^(a), f₂ ^(b), f₃ ^(b) by applying two pump tones at frequencies f₁₂^(p)=f₁ ^(a)+f₂ ^(b) and f₁₃ ^(p)=f₁ ^(a)+f₃ ^(b) sequentially. Thus,using this method of entanglement, the microwave quantum device 1000 isconfigured to remotely entangle n different qubits by applyingsequentially n−1 pumps that satisfy the sum condition of the readoutfrequencies of those qubits.

Generation of Multiple Pairs of Entangled Photons

The generation of multiple pairs of entangled photons is a more generalapplication of the multimode JPC 130 than the remote entanglement ofmultiple qubits by applying entangled photons to readout their state.

In FIG. 10, instead of the bus_a 810A being connected to the readoutresonators 802 and qubits 820A, the bus_a 810A may be connected to afirst quantum system (not shown) to receive entangled photons. Also, inFIG. 10, instead of the bus_b 810B being connected to the readoutresonators 804 and qubits 820B, the bus_b 810B may be connected to asecond quantum system (not shown) to receive entangled photons.

In another implementation, a first quantum system is connected directlyto the circulator 1015A without bus_a 810A and a second quantum systemis connected directly to the circulator 1015B without bus_b 810B.

Another beneficial property of the amplification process taking place inthe multimode JPC 130 is that the multimode JPC 130 creates a two-modesqueezed state between the two nondegenerate modes of the JPC 130,according to an embodiment. In other words, the pump photons of the pump(P), which supply the energy for the amplification process, aredown-converted via interaction with the dispersive nonlinear medium(i.e., JRM 105) in the multimode JPC 130; via the application of pumptones to the pump (P) port, the multimode JPC 130 generates multiplepairs of entangled photons whose frequencies lie within the dynamicalbandwidths of the nondegenerate resonance modes of the multimode JPC130, and whose frequency sum equals to the frequency of the applied pumptone.

Utilizing this property in a multimode nondegenerate device (i.e., themultimode JPC 130) allows for a wide range of applications in quantuminformation processing, such as generation of multiple entangled pairsof photons corresponding to different resonance frequencies of themultimode JPC 130 (particularly of the multimode resonators 115A and115B) by applying simultaneously or consequentially multiple pump tones.

FIG. 11 is flow chart 1100 of a method of operating a multimodeJosephson parametric converter 130 to generate multiple pairs ofentangled photons according to an embodiment.

At block 1105, the first multimode resonator_a 115A in the multimodeJosephson parametric converter 130 is configured to receive a firstgroup of signals at different resonance frequencies f₁ ^(a), f₂ ^(a), f₃^(a), . . . , f_(n) ^(a) in which the first group of signals correspondsto resonance modes of the first multimode resonator_a 115A, where thefirst multimode resonator is a first left-handed transmission line.

At block 1110, the second multimode resonator_b 115B in the multimodeJosephson parametric converter 130 is configured to receive a secondgroup of signals at different resonance frequencies f₁ ^(b), f₂ ^(b), f₃^(b), . . . , f_(n) ^(b) in which the second group of signalscorresponds to resonance modes of the second multimode resonator_b 115B,where the second multimode resonator is a second left-handedtransmission line.

At block 1115, pump signals are received by the second multimoderesonator_b 115B, where the pump signals are a first (pump) frequencysum, a second (pump) frequency sum, through a last (pump) frequency sum.It is noted that although the pump is fed through the feedlines ofresonator_b 115B and not through the feedlines of resonator_a 115A inFIG. 3, the pump may be injected through either sigma (Σ) port of the180 hybrids. Accordingly, the 50Ω termination can be moved to the otherside. Also, it is noted that the pumps do not need be applied inparallel, which allows for switching between different entangled pairssequentially or in a certain order

At block 1120, the multimode Josephson parametric converter 130 isconfigured to generate simultaneously or sequentially pairs of entangledphotons, and the pairs of entangled photons include a first photon pair,a second photon pair, through a last photon pair.

For example, one photon in the first entangled photon pair may exit theJPC 130 via port_a 120A of the multimode resonator_a 115A at a readoutfrequency corresponding to a particular readout resonator 802 (e.g.,readout resonator 802_1) and be transmitted to the circulator 1015A. Thecirculator 1015A may transmit the signal having the one photon to thebus_a 810A at a readout frequency that causes one of the qubits (e.g.,qubit q₁ ^(a)) to be readout by the readout resonator 802_1.Concurrently, the other entangled photon in the first photon pair mayexit the JPC 130 via port_b 120B of the multimode resonator_b 115B at areadout frequency corresponding to a particular readout resonator 804(e.g., readout resonator 804_1) and be transmitted to the circulator1015B. The circulator 1015B may transmit the signal having the otherphoton to the bus_b 810B at a readout frequency that causes one of thequbits (e.g., qubit q₁ ^(b)) to be read out by the readout resonator804_1. The initial entanglement of the photons in the first photon pairare utilized to cause the entanglement of qubit pair (q₁ ^(a), q₁ ^(b)).In another implementation, instead of outputting the entangled photonsof the first photon pair from resonators 115A, 115B to readoutresonators 802, 804, the photons of the first photon pair may berespectively output to a first and second quantum system, respectively.In this case, the entangled photons of the first photon pair stillentangle aspects of the separate first and second quantum systems.

The first frequency sum is a summation of one resonance frequency of thefirst group plus one resonance frequency of the second group. The secondfrequency sum is a summation of another resonance frequency of the firstgroup plus another resonance frequency of the second group. The thirdfrequency sum is a summation of yet another resonance frequency of thefirst group plus yet another resonance frequency of the second group.

The first pair of the entangled photons corresponds to a first photon atthe one resonance frequency of the first group (having been received bya quantum system connected to the multimode resonator_a 115A) and afirst photon at the one resonance frequency of the second group (havingbeen received by another quantum system connected to the multimoderesonator_b 115B), in response to energy of the first frequency sum ofthe pump drive applied to the JPC 130 being down-converted viainteraction with a dispersive nonlinear medium (e.g., JRM 105).Similarly, the second pair of the entangled photons corresponds to asecond photon at the another resonance frequency of the first group anda second photon at the another resonance frequency of the second group,in response to energy of the second frequency sum of the pump drivebeing down-converted via interaction with the dispersive nonlinearmedium. Likewise, the third pair of the entangled photons corresponds toa third photon at the yet another resonance frequency of the first groupand a third photon at the yet another resonance frequency of the secondgroup, in response to energy of the third frequency sum of the pumpdrive being down-converted via interaction with the dispersive nonlinearmedium. As noted herein, certain combinations are pointed out for easeof understanding, but embodiments are not limited to the examplecombinations. It is understood that there are many more combinations.

The first group of signals and the second group of signals at thedifferent resonance frequencies range from about 5-15 GHz.

Bell State Generator

Referring to FIGS. 1-6, this embodiment is useful in quantum computationschemes that use photons as quantum bits and utilizes the multimode JPC130 that is comprised of two resonators (i.e., multimode resonator_a115A and multimode resonator_b 115B) that are spatially nondegenerate(have different spatial ports) but temporally degenerate (have the sameresonance frequencies). In particular, the multimode JPC 130 isconfigured such that two resonance modes in multimode resonator_a 115Ahave resonance frequencies that coincide with two resonance modes inmultimode resonator_b 115B, i.e., f₁ ^(a)=f₁ ^(b)=f₁ and f₂ ^(a)=f₂^(b)=f₂. Also, it is noted that in this technique the JPC 130 does notfunction as an amplifier, but rather as a single-photon downconveter;hence, certain parameters of the device are modified and redesignedaccordingly. In particular, the JRM and the coupling to the left-handedresonators are engineered such that a single pump photon getsdownconverted to one pair of signal and idler photons. This canpartially be achieved by using small Josephson junctions with smallcritical current in the JRM, and making the pump ‘soft’ (applied, e.g.,at resonance).

By applying a pump tone at f^(p)=f₁ ^(a)+f₂ ^(b)=f₂ ^(a)+f₁ ^(b)=f₁+f₂to the multimode JPC 130, photons that get generated as a result of thedown conversion process of pump photons via nonlinear interaction withthe JRM 105 of the multimode JPC 130 are in an equal superposition ofspatial states (in multimode resonator_a 115A and multimode resonator_b115B).

The following example is provided. If experimenters assume without lossof generality that f₂>f₁ and denote by the quantum states |0^(f1) _(a)

and |0^(f1) _(b)

the lack of a photon in multimode resonator_a 115A and multimoderesonator_b 115B (depending on the subscript) at frequency f1 and by thequantum states |0^(f2) _(a)

and |0^(f2) _(b)

the lack of a photon in multimode resonator_a 115A and multimoderesonator_b 115B (depending on the subscript) at frequency f2, andsimilarly denote by the quantum states |1^(f1) _(a)

and |1^(f1) _(b)

the presence of a photon in multimode resonator_a 115A and multimoderesonator_b 115B (depending on the subscript) at frequency f1 and by thequantum states |1^(f2) _(a)

and |1^(f2) _(b)

the presence of a photon in multimode resonator_a 115A and multimoderesonator_b 115B (depending on the subscript) at frequency f2, then thequantum state of the system of the multimode JPC 130 can be written as:

$\left. {b_{2}b_{1}a_{2}a_{1}} \right\rangle = {\frac{1}{\sqrt{2}}{\left( {\left. {1_{b}0_{b}0_{a}1_{a}} \right\rangle + \left. {0_{b}1_{b}1_{a\;}0_{a}} \right\rangle} \right).}}$

In the above equation, the subscript “a” denotes multimode resonator_a115A, and the subscript “b” denotes multimode resonator_b 115B. Theparameter (or place holder) “a1” which can have a value “0” or “1”represents the lack or presence of a photon at frequency f1 in multimoderesonator_a 115A respectively. The parameter (or place holder) “a2”which can have a value “0” or “1” represents the lack or presence of aphoton at frequency f2 in multimode resonator_a 115A respectively. Theparameter (or place holder) “b1” which can have a value “0” or “1”represents the lack or presence of a photon at frequency f1 in multimoderesonator_b 115B respectively. The parameter (or place holder) “b2”which can have a value “0” or “1” represents the lack or presence of aphoton at frequency f2 in multimode resonator_b 115B.

Furthermore, by associating logical qubit states with photons beingeither in mode 1 or 2 of resonators 115A and 115B, i.e., |0_(a)

_(L)=|0^(f2) _(a1) ^(f1) _(a), |0_(b)

_(L)=|0^(f2) _(b)1^(f1) _(b)

, |1_(a)

_(L)=|1^(f2) _(a)0^(f1) _(a)

and |1_(b)

_(L)=|1^(f2) _(b)0^(f1) _(b)

, then the quantum state of the system for the JPC 130 can be rewrittenas

${\left. {b_{L}a_{L}} \right\rangle = {\frac{1}{\sqrt{2}}\left( {\left. {1_{b}0_{a}} \right\rangle_{L} + \left. {0_{b}1_{a}} \right\rangle_{L}} \right)}},$

which represents a Bell state for logical qubits ‘a’ and ‘b’ whosecomputation state ‘0a,b’ corresponds to the case where a photon ispresent in mode 1 and lacking in mode 2 of multimode resonator_a andmultimode resonator_b respectively, and whose computation state ‘1 a,b’corresponds to the case where a photon is lacking in mode 1 and presentin mode 2 of multimode resonator_a and multimode resonator_brespectively.

FIG. 12 is a flow chart 1200 of a method of generating a bell stateusing photons as quantum bits according to an embodiment.

At block 1205, a first multimode resonator_a 115A and a second multimoderesonator_b 115B (in JPC 130) are both connected to a dispersivenonlinear medium (e.g., JRM 105), where the first multimode resonator isa first left-handed transmission line and the second multimode resonatoris a second left-handed transmission line, and where resonance modes areidentical in the first and second multimode resonators (i.e., f₁ ^(a),f₂ ^(a), f₃ ^(a), . . . , f_(n) ^(a) respectively equal f₁ ^(b), f₂^(b), f₃ ^(b), . . . , f_(n) ^(b)).

At block 1210, the second multimode resonator_b 115B receives a pumpsignal at a frequency sum, the frequency sum is a summation of a(certain) resonance frequency of the resonance modes plus anotherresonance frequency of the resonance modes.

At block 1215, a first photon and a second photon in an equalsuperposition of spatial states are generated in the JPC 130, where theequal superposition of the spatial states for the first and secondphotons are related to being in the first multimode resonator_a 115A andthe second multimode resonator_b 115B.

The equal superposition of the spatial states for the first photon isconfigured such that the first photon has an equal probability of beingin (or not being in) the first multimode resonator_a 115A or the secondmultimode resonator_b 115B. The equal superposition of the spatialstates for the second photon is configured such that the second photonhas an equal probability of being in (or not being in) the firstmultimode resonator_a 115A or the second multimode resonator_b 115B.

The first photon and the second photon are not both in the same one ofthe first and second multimode resonators 115A, 115B. That is, if thefirst photon is in the first multimode resonator_a 115A, then the secondphoton is in the second multimode resonator_b 115B. Conversely, if thesecond photon is in the first multimode resonator_a 115A, then the firstphoton is in the second multimode resonator_b 115B.

Unitary Frequency Conversion Between Two Propagating Microwave Signalsthat are Close in Frequency

The feature that the density of modes (frequency resonance modes) of JPCresonators (multimode resonator_a 115A and multimode resonator_b 115B)which are comprised of left-handed transmission lines is large close totheir low-frequency bound f_(IR)=ω_(IR)/2π allows the multimode JPC 130to perform unitary frequency conversion between pairs of propagatingmicrowave modes that are close in frequency (e.g., tens of megahertz) bypumping (via pumps P) the multimode JPC 130 with parallel pump toneswhose frequencies correspond to the frequency difference between theresonance frequencies (of the propagating microwave signals (e.g., Ssignal and I signal to respective resonators 115A and 115B).

For example, if a quantum signal (e.g., S) at frequency f₁ ^(a) is sentto the multimode resonator_a 115A and a quantum signal (e.g., I) atfrequency f₂ ^(b) is sent to the multimode resonator_b 115B, in order toperform a unitary frequency conversion between this pair of propagatingmicrowave modes pump tone at the frequency difference f₁ ^(p)=|f₁^(a)−f₂ ^(b)| is injected to the JPC 130. The term unitary frequencyconversion means that the signal is converted from one frequency toanother without loss of information, or in other words the signal isconverted in a lossless (no photons are lost) and coherent (the phase ispreserved) manner. Analogously, this conversion process can begeneralized to other pairs of propagating microwave signals input onport_a and port_b and whose frequencies lie within the bandwidths ofresonance modes of resonator_a and resonator_b that couple to the JRM byapplying appropriate pump drives.

According to one implementation, if the pump frequencies required toperform the frequency conversion between the modes lie below the cutofffrequency of the metamaterial resonators 115A and 115B, a particularmethod of pumping the JRM 105 may be applied that is different frominjecting the pump tones directly to the multimode resonators 115A and115B. One such method is pumping the JRM 105 using a three-port powerdivider capacitively coupled to the JRM 105 directly.

It will be noted that various microelectronic device fabrication methodsmay be utilized to fabricate the components/elements discussed herein asunderstood by one skilled in the art. In semiconductor orsuperconducting device fabrication, the various processing steps fallinto four general categories: deposition, removal, patterning, andmodification of electrical properties.

Deposition is any process that grows, coats, or otherwise transfers amaterial onto the wafer. Available technologies include physical vapordeposition (PVD), chemical vapor deposition (CVD), electrochemicaldeposition (ECD), molecular beam epitaxy (MBE) and more recently, atomiclayer deposition (ALD) among others.

Removal is any process that removes material from the wafer: examplesinclude etch processes (either wet or dry), and chemical-mechanicalplanarization (CMP), etc.

Patterning is the shaping or altering of deposited materials, and isgenerally referred to as lithography. For example, in conventionallithography, the wafer is coated with a chemical called a photoresist;then, a machine called a stepper focuses, aligns, and moves a mask,exposing select portions of the wafer below to short wavelength light;the exposed regions are washed away by a developer solution. Afteretching or other processing, the remaining photoresist is removed.Patterning also includes electron-beam lithography.

Modification of electrical properties may include doping, such as dopingtransistor sources and drains, generally by diffusion and/or by ionimplantation. These doping processes are followed by furnace annealingor by rapid thermal annealing (RTA). Annealing serves to activate theimplanted dopants.

The flowchart and block diagrams in the Figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods, and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams may represent a module, segment, or portionof instructions, which comprises one or more executable instructions forimplementing the specified logical function(s). In some alternativeimplementations, the functions noted in the block may occur out of theorder noted in the figures. For example, two blocks shown in successionmay, in fact, be executed substantially concurrently, or the blocks maysometimes be executed in the reverse order, depending upon thefunctionality involved. It will also be noted that each block of theblock diagrams and/or flowchart illustration, and combinations of blocksin the block diagrams and/or flowchart illustration, can be implementedby special purpose hardware-based systems that perform the specifiedfunctions or acts or carry out combinations of special purpose hardwareand computer instructions.

The descriptions of the various embodiments of the present inventionhave been presented for purposes of illustration, but are not intendedto be exhaustive or limited to the embodiments disclosed. Manymodifications and variations will be apparent to those of ordinary skillin the art without departing from the scope and spirit of the describedembodiments. The terminology used herein was chosen to best explain theprinciples of the embodiments, the practical application or technicalimprovement over technologies found in the marketplace, or to enableothers of ordinary skill in the art to understand the embodimentsdisclosed herein.

What is claimed is:
 1. A method of generating a bell state using photonsas quantum bits, the method comprising: providing a first multimoderesonator and a second multimode resonator both connected to adispersive nonlinear medium, wherein the first multimode resonator is afirst left-handed transmission line and the second multimode resonatoris a second left-handed transmission line, wherein resonance modes areidentical in the first and second multimode resonators; receiving, bythe second multimode resonator, a pump signal at a frequency sum, thefrequency sum is a summation of a resonance frequency of the resonancemodes plus another resonance frequency of the resonance modes; and inresponse to receiving the pump signal at the frequency sum, generating,by the dispersive nonlinear medium, a first photon and a second photonin a superposition of location states, the superposition of the locationstates for the first and second photons are related to being in thefirst multimode resonator and the second multimode resonator.
 2. Themethod of claim 1, wherein the superposition of the location states forthe first photon is configured such that the first photon has an equalprobability of being in the first multimode resonator or the secondmultimode resonator.
 3. The method of claim 2, wherein the superpositionof the location states for the second photon is configured such that thesecond photon has an equal probability of being in the first multimoderesonator or the second multimode resonator.
 4. The method of claim 3,wherein the first photon and the second photon are not both in a sameone of the first and second multimode resonators.
 5. The method of claim1, wherein the superposition of the location states for the first photonis configured such that the first photon has an equal probability of notbeing in the first multimode resonator or the second multimoderesonator.
 6. The method of claim 5, wherein the superposition of thelocation states for the second photon is configured such that the secondphoton has an equal probability of not being in the first multimoderesonator or the second multimode resonator.
 7. The method of claim 1,wherein the resonance modes of the first and second multimode resonatorscomprise a first resonance frequency, a second resonance frequency,through a last resonance frequency.
 8. The method of claim 7, whereinthe resonance frequency is one of the first resonance frequency throughthe last resonance frequency.
 9. The method of claim 8, wherein theanother resonance frequency is another one of the first resonancefrequency through the last resonance frequency.
 10. The method of claim1, wherein a multimode Josephson parametric converter is formed by thefirst multimode resonator, the second multimode resonator, and thedispersive nonlinear medium.
 11. The method of claim 10, wherein themultimode Josephson parametric converter is not configured to functionas an amplifier.
 12. The method of claim 10, wherein the multimodeJosephson parametric converter is configured to function as asingle-photon downconverter.
 13. The method of claim 12, wherein thepump signal at the frequency sum is down converted into the first photonand the second photon.
 14. The method of claim 10, wherein thedispersive nonlinear medium is a multimode Josephson ring modulator. 15.The method of claim 14, wherein the multimode Josephson ring modulatorcomprises Josephson junctions.
 16. The method of claim 15, wherein theJosephson junctions are connected in a ring.
 17. The method of claim 10,wherein the multimode Josephson parametric converter comprises a firstport and a second port.
 18. The method of claim 17, wherein the firstport or the second port is configured to receive input of the pumpsignal at the frequency sum.
 19. The method of claim 17, wherein thefirst port and the second port are each connectable to a 180 degreehybrid coupler.
 20. The method of claim 1, wherein the resonance modesrange from about 5-15 GHz.